This talk presents a study of Cauchy's additive equation f(x + y) = f(x) + f(y) and the related equations. All continuous solutions to the additive equation on R are linear. We seek other solutions to the additive equation, assuming weaker conditions on f. Our goal is to find necessary and sufficient conditions on an additive function f which make f linear. For example, we say that f is {\it locally bounded} if there exits an interval [a,b] and a bound M such that f(x) \leq M or f(x) \geq M for all x \in [a,b]. We show that if f is additive and locally bounded, then f is linear. We then use the idea of locally bounded to show that if f is additive and locally Lebesgue integrable, then f is linear. We refer to these results as our analytic results.

These results would lead one to believe that all additive functions on R are linear. This, however, is not true. To construct these non-linear solutions, we need a {\it Hamel basis} for R over Q, which is a set $\set{ {\beta}_{\alpha} }_{\alpha \in A} \subset \R$ which is linearly independent over Q and has the property that every element of R can be written as a finite linear combination of elements of $\set{ {\beta}_{\alpha} }$. Given a Hamel basis, we construct a non-linear additive function f relative this basis. We then show that the graph { (x, f(x) } of this function is quite interesting -- the graph is a dense set in R}^2. We conclude our talk by showing how the analytic results can be related to corresponding results given in terms of a Hamel basis, and relating these notions to generalizations of measure.